<!DOCTYPE html>
<html class="client-nojs" lang="en" dir="ltr" xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
    <head>
        <meta charset="UTF-8" />
        <title>
            Hermitian matrix - Wikipedia
        </title>
        <script>
        <![CDATA[
        document.documentElement.className="client-js";RLCONF={"wgBreakFrames":!1,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgMonthNamesShort":["","Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"],"wgRequestId":"XlQ4hQpAMFMAAF2Dp-oAAABV","wgCSPNonce":!1,"wgCanonicalNamespace":"","wgCanonicalSpecialPageName":!1,"wgNamespaceNumber":0,"wgPageName":"Hermitian_matrix","wgTitle":"Hermitian matrix","wgCurRevisionId":942460710,"wgRevisionId":942460710,"wgArticleId":189682,"wgIsArticle":!0,"wgIsRedirect":!1,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Use American English from January 2019","All Wikipedia articles written in American English","Articles with short description","Articles to be expanded from February 2018","All articles to be expanded",
        "All articles with unsourced statements","Articles with unsourced statements from September 2019","Matrices"],"wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Hermitian_matrix","wgRelevantArticleId":189682,"wgIsProbablyEditable":!0,"wgRelevantPageIsProbablyEditable":!0,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgMediaViewerOnClick":!0,"wgMediaViewerEnabledByDefault":!0,"wgPopupsReferencePreviews":!1,"wgPopupsConflictsWithNavPopupGadget":!1,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":!0,"nearby":!0,"watchlist":!0,"tagline":!1},"wgWMESchemaEditAttemptStepOversample":!1,"wgULSCurrentAutonym":"English","wgNoticeProject":"wikipedia","wgWikibaseItemId":"Q652941","wgCentralAuthMobileDomain":!1,"wgEditSubmitButtonLabelPublish":!0};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","noscript":"ready","user.styles":
        "ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"ready","user.tokens":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","mediawiki.legacy.shared":"ready","mediawiki.legacy.commonPrint":"ready","mediawiki.toc.styles":"ready","skins.vector.styles":"ready","wikibase.client.init":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.math.scripts","ext.cite.ux-enhancements","site","mediawiki.page.startup","skins.vector.js","mediawiki.page.ready","mediawiki.toc","ext.gadget.ReferenceTooltips","ext.gadget.watchlist-notice","ext.gadget.DRN-wizard","ext.gadget.charinsert","ext.gadget.refToolbar","ext.gadget.extra-toolbar-buttons","ext.gadget.switcher","ext.centralauth.centralautologin","mmv.head","mmv.bootstrap.autostart","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.eventLogging","ext.wikimediaEvents",
        "ext.navigationTiming","ext.uls.compactlinks","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.quicksurveys.init","ext.centralNotice.geoIP","ext.centralNotice.startUp"];
        ]]>
        </script>
        <script>
        <![CDATA[
        (RLQ=window.RLQ||[]).push(function(){mw.loader.implement("user.tokens@tffin",function($,jQuery,require,module){/*@nomin*/mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"});
        });});
        ]]>
        </script>
        <link rel="stylesheet" href="/w/load.php?lang=en&amp;modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cmediawiki.legacy.commonPrint%2Cshared%7Cmediawiki.toc.styles%7Cskins.vector.styles%7Cwikibase.client.init&amp;only=styles&amp;skin=vector" />
        <script async="async" src="/w/load.php?lang=en&amp;modules=startup&amp;only=scripts&amp;raw=1&amp;skin=vector"></script>
        <meta name="ResourceLoaderDynamicStyles" content="" />
        <link rel="stylesheet" href="/w/load.php?lang=en&amp;modules=site.styles&amp;only=styles&amp;skin=vector" />
        <meta name="generator" content="MediaWiki 1.35.0-wmf.20" />
        <meta name="referrer" content="origin" />
        <meta name="referrer" content="origin-when-crossorigin" />
        <meta name="referrer" content="origin-when-cross-origin" />
        <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Hermitian_matrix&amp;action=edit" />
        <link rel="edit" title="Edit this page" href="/w/index.php?title=Hermitian_matrix&amp;action=edit" />
        <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png" />
        <link rel="shortcut icon" href="/static/favicon/wikipedia.ico" />
        <link rel="search" type="application/opensearchdescription+xml" href="/w/opensearch_desc.php" title="Wikipedia (en)" />
        <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd" />
        <link rel="license" href="//creativecommons.org/licenses/by-sa/3.0/" />
        <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&amp;feed=atom" />
        <link rel="canonical" href="https://en.wikipedia.org/wiki/Hermitian_matrix" />
        <link rel="dns-prefetch" href="//login.wikimedia.org" />
        <link rel="dns-prefetch" href="//meta.wikimedia.org" /><!--[if lt IE 9]><script src="/w/resources/lib/html5shiv/html5shiv.js"></script><![endif]-->

        <style data-mw-deduplicate="TemplateStyles:r935243608">
        <![CDATA[
        .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
        ]]>
        </style>
    </head>
    <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Hermitian_matrix rootpage-Hermitian_matrix skin-vector action-view">
        <div id="mw-page-base" class="noprint"></div>
        <div id="mw-head-base" class="noprint"></div>
        <div id="content" class="mw-body" role="main">
            <a id="top"></a>
            <div id="siteNotice" class="mw-body-content">
                <!-- CentralNotice -->
            </div>
            <div class="mw-indicators mw-body-content"></div>
            <h1 id="firstHeading" class="firstHeading" lang="en" xml:lang="en">
                Hermitian matrix
            </h1>
            <div id="bodyContent" class="mw-body-content">
                <div id="siteSub" class="noprint">
                    From Wikipedia, the free encyclopedia
                </div>
                <div id="contentSub"></div>
                <div id="jump-to-nav"></div><a class="mw-jump-link" href="#mw-head">Jump to navigation</a> <a class="mw-jump-link" href="#p-search">Jump to search</a>
                <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr" xml:lang="en">
                    <div class="mw-parser-output">
                        <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">
                            Matrix equal to its conjugate-transpose
                        </div>
                        <div role="note" class="hatnote navigation-not-searchable">
                            For matrices with symmetry over the <a href="/wiki/Real_number" title="Real number">real number</a> field, see <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>.
                        </div>
                        <p>
                            In mathematics, a <b>Hermitian matrix</b> (or <b>self-adjoint matrix</b>) is a <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> that is equal to its own <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>—that is, the element in the <span class="texhtml mvar" style="font-style:italic;">i</span>-th row and <span class="texhtml mvar" style="font-style:italic;">j</span>-th column is equal to the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the element in the <span class="texhtml mvar" style="font-style:italic;">j</span>-th row and <span class="texhtml mvar" style="font-style:italic;">i</span>-th column, for all indices <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml mvar" style="font-style:italic;">j</span>:
                        </p>
                        <div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; background-color: #F5FFFA; text-align: center; display: table">
                            <p>
                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mtext>
                                                    &nbsp;Hermitian
                                                </mtext>
                                            </mrow>
                                            <mspace width="1em"></mspace>
                                            <mspace width="thickmathspace"></mspace>
                                            <mo stretchy="false">
                                                ⟺<!-- ⟺ -->
                                            </mo>
                                            <mspace width="thickmathspace"></mspace>
                                            <mspace width="1em"></mspace>
                                            <msub>
                                                <mi>
                                                    a
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <msub>
                                                        <mi>
                                                            a
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                            <mi>
                                                                i
                                                            </mi>
                                                        </mrow>
                                                    </msub>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0aaa74b2267a48312e19321211cd9e3a39228" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:32.77ex; height:3.009ex;" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}" /></span>
                            </p>
                        </div>
                        <p>
                            or in matrix form:
                        </p>
                        <dl>
                            <dd>
                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mtext>
                                                    &nbsp;Hermitian
                                                </mtext>
                                            </mrow>
                                            <mspace width="1em"></mspace>
                                            <mspace width="thickmathspace"></mspace>
                                            <mo stretchy="false">
                                                ⟺<!-- ⟺ -->
                                            </mo>
                                            <mspace width="thickmathspace"></mspace>
                                            <mspace width="1em"></mspace>
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <msup>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mi mathvariant="sans-serif">
                                                                    T
                                                                </mi>
                                                            </mrow>
                                                        </mrow>
                                                    </msup>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca00b61ff0e264e6c1e5adc9a00c0d2751feecf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:32.194ex; height:3.509ex;" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}" /></span>.
                            </dd>
                        </dl>
                        <p>
                            Hermitian matrices can be understood as the complex extension of real <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a>.
                        </p>
                        <p>
                            If the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of a matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {H}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        H
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A^{\mathsf {H}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.139ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {H}}}" /></span>, then the Hermitian property can be written concisely as
                        </p>
                        <div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; background-color: #F5FFFA; text-align: center; display: table">
                            <p>
                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mtext>
                                                    &nbsp;Hermitian
                                                </mtext>
                                            </mrow>
                                            <mspace width="1em"></mspace>
                                            <mspace width="thickmathspace"></mspace>
                                            <mo stretchy="false">
                                                ⟺<!-- ⟺ -->
                                            </mo>
                                            <mspace width="thickmathspace"></mspace>
                                            <mspace width="1em"></mspace>
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291d260bf69b764e75818669ab27870d58771e1f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:32.123ex; height:2.676ex;" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}" /></span>
                            </p>
                        </div>
                        <p>
                            Hermitian matrices are named after <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>. Other, equivalent notations in common use are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        H
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                        <mo>
                                            =
                                        </mo>
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo>
                                                    †<!-- † -->
                                                </mo>
                                            </mrow>
                                        </msup>
                                        <mo>
                                            =
                                        </mo>
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo>
                                                    ∗<!-- ∗ -->
                                                </mo>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa270391d183478251283d2c4b2c72ac4563352" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:14.839ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}" /></span>, although note that in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\ast }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo>
                                                    ∗<!-- ∗ -->
                                                </mo>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A^{\ast }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5541bfa07743be995242c892c344395e672d6fa2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="A^{\ast }" /></span> typically means the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> only, and not the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>.
                        </p>
                        <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading">
                            <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" />
                            <div class="toctitle" lang="en" dir="ltr" xml:lang="en">
                                <h2 id="mw-toc-heading">
                                    Contents
                                </h2>
                            </div>
                            <ul>
                                <li class="toclevel-1 tocsection-1">
                                    <a href="#Alternative_characterizations"><span class="tocnumber">1</span> <span class="toctext">Alternative characterizations</span></a>
                                    <ul>
                                        <li class="toclevel-2 tocsection-2">
                                            <a href="#Equality_with_the_adjoint"><span class="tocnumber">1.1</span> <span class="toctext">Equality with the adjoint</span></a>
                                        </li>
                                        <li class="toclevel-2 tocsection-3">
                                            <a href="#Reality_of_quadratic_forms"><span class="tocnumber">1.2</span> <span class="toctext">Reality of quadratic forms</span></a>
                                        </li>
                                        <li class="toclevel-2 tocsection-4">
                                            <a href="#Spectral_properties"><span class="tocnumber">1.3</span> <span class="toctext">Spectral properties</span></a>
                                        </li>
                                    </ul>
                                </li>
                                <li class="toclevel-1 tocsection-5">
                                    <a href="#Applications"><span class="tocnumber">2</span> <span class="toctext">Applications</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-6">
                                    <a href="#Examples"><span class="tocnumber">3</span> <span class="toctext">Examples</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-7">
                                    <a href="#Properties"><span class="tocnumber">4</span> <span class="toctext">Properties</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-8">
                                    <a href="#Decomposition_into_Hermitian_and_skew-Hermitian"><span class="tocnumber">5</span> <span class="toctext">Decomposition into Hermitian and skew-Hermitian</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-9">
                                    <a href="#Rayleigh_quotient"><span class="tocnumber">6</span> <span class="toctext">Rayleigh quotient</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-10">
                                    <a href="#See_also"><span class="tocnumber">7</span> <span class="toctext">See also</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-11">
                                    <a href="#References"><span class="tocnumber">8</span> <span class="toctext">References</span></a>
                                </li>
                                <li class="toclevel-1 tocsection-12">
                                    <a href="#External_links"><span class="tocnumber">9</span> <span class="toctext">External links</span></a>
                                </li>
                            </ul>
                        </div>
                        <h2>
                            <span class="mw-headline" id="Alternative_characterizations">Alternative characterizations</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=1" title="Edit section: Alternative characterizations">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <p>
                            Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
                        </p>
                        <h3>
                            <span class="mw-headline" id="Equality_with_the_adjoint">Equality with the adjoint</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=2" title="Edit section: Equality with the adjoint">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h3>
                        <p>
                            A square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is Hermitian if and only if it is equal to its <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">adjoint</a>, that is, it satisfies
                        </p>
                        <div class="mwe-math-element">
                            <div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;">
                                <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mo fence="false" stretchy="false">
                                                ⟨<!-- ⟨ -->
                                            </mo>
                                            <mi>
                                                w
                                            </mi>
                                            <mo>
                                                ,
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                v
                                            </mi>
                                            <mo fence="false" stretchy="false">
                                                ⟩<!-- ⟩ -->
                                            </mo>
                                            <mo>
                                                =
                                            </mo>
                                            <mo fence="false" stretchy="false">
                                                ⟨<!-- ⟨ -->
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                w
                                            </mi>
                                            <mo>
                                                ,
                                            </mo>
                                            <mi>
                                                v
                                            </mi>
                                            <mo fence="false" stretchy="false">
                                                ⟩<!-- ⟩ -->
                                            </mo>
                                            <mo>
                                                ,
                                            </mo>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}
                                    </annotation>
                                </semantics></math>
                            </div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459de45e76bace9d04a80d2e8efc2abbbc246047" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:18.501ex; height:2.843ex;" alt="{\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}" />
                        </div>for any pair of vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,w}">
                        <semantics>
                            <mrow class="MJX-TeXAtom-ORD">
                                <mstyle displaystyle="true" scriptlevel="0">
                                    <mi>
                                        v
                                    </mi>
                                    <mo>
                                        ,
                                    </mo>
                                    <mi>
                                        w
                                    </mi>
                                </mstyle>
                            </mrow>
                            <annotation encoding="application/x-tex">
                                {\displaystyle v,w}
                            </annotation>
                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.009ex;" alt="v,w" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">
                        <semantics>
                            <mrow class="MJX-TeXAtom-ORD">
                                <mstyle displaystyle="true" scriptlevel="0">
                                    <mo fence="false" stretchy="false">
                                        ⟨<!-- ⟨ -->
                                    </mo>
                                    <mo>
                                        ⋅<!-- ⋅ -->
                                    </mo>
                                    <mo>
                                        ,
                                    </mo>
                                    <mo>
                                        ⋅<!-- ⋅ -->
                                    </mo>
                                    <mo fence="false" stretchy="false">
                                        ⟩<!-- ⟩ -->
                                    </mo>
                                </mstyle>
                            </mrow>
                            <annotation encoding="application/x-tex">
                                {\displaystyle \langle \cdot ,\cdot \rangle }
                            </annotation>
                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" /></span> denotes <a href="/wiki/Dot_product" title="Dot product">the inner product</a> operation.
                        <p>
                            This is also the way that the more general concept of <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a> is defined.
                        </p>
                        <h3>
                            <span class="mw-headline" id="Reality_of_quadratic_forms">Reality of quadratic forms</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=3" title="Edit section: Reality of quadratic forms">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h3>
                        <p>
                            A square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is Hermitian if and only if it is such that
                        </p>
                        <div class="mwe-math-element">
                            <div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;">
                                <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mo fence="false" stretchy="false">
                                                ⟨<!-- ⟨ -->
                                            </mo>
                                            <mi>
                                                v
                                            </mi>
                                            <mo>
                                                ,
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                v
                                            </mi>
                                            <mo fence="false" stretchy="false">
                                                ⟩<!-- ⟩ -->
                                            </mo>
                                            <mo>
                                                ∈<!-- ∈ -->
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mi mathvariant="double-struck">
                                                    R
                                                </mi>
                                            </mrow>
                                            <mo>
                                                ,
                                            </mo>
                                            <mspace width="1em"></mspace>
                                            <mi>
                                                v
                                            </mi>
                                            <mo>
                                                ∈<!-- ∈ -->
                                            </mo>
                                            <mi>
                                                V
                                            </mi>
                                            <mo>
                                                .
                                            </mo>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}
                                    </annotation>
                                </semantics></math>
                            </div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997ea0350c18735926412de88420ac9ca989f50c" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:21.119ex; height:2.843ex;" alt="{\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}" />
                        </div>
                        <h3>
                            <span class="mw-headline" id="Spectral_properties">Spectral properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=4" title="Edit section: Spectral properties">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h3>
                        <p>
                            A square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is Hermitian if and only if it is unitarily <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a> with real <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>.
                        </p>
                        <h2>
                            <span class="mw-headline" id="Applications">Applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=5" title="Edit section: Applications">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <p>
                            Hermitian matrices are fundamental to the quantum theory of <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> created by <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, and <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> in 1925.
                        </p>
                        <h2>
                            <span class="mw-headline" id="Examples">Examples</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=6" title="Edit section: Examples">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <p>
                            In this section, the conjugate transpose of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {H}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        H
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A^{\mathsf {H}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.139ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {H}}}" /></span>, the transpose of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {T}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        T
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A^{\mathsf {T}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bf0331204e30cba9ec7f695dfea97e6745a7c2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {T}}}" /></span> and conjugate of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mrow class="MJX-TeXAtom-ORD">
                                            <mover>
                                                <mi>
                                                    A
                                                </mi>
                                                <mo accent="false">
                                                    ¯<!-- ¯ -->
                                                </mo>
                                            </mover>
                                        </mrow>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle {\overline {A}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}" /></span>.
                        </p>
                        <p>
                            See the following example:
                        </p>
                        <dl>
                            <dd>
                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow>
                                                    <mo>
                                                        [
                                                    </mo>
                                                    <mtable rowspacing="4pt" columnspacing="1em">
                                                        <mtr>
                                                            <mtd>
                                                                <mn>
                                                                    2
                                                                </mn>
                                                            </mtd>
                                                            <mtd>
                                                                <mn>
                                                                    2
                                                                </mn>
                                                                <mo>
                                                                    +
                                                                </mo>
                                                                <mi>
                                                                    i
                                                                </mi>
                                                            </mtd>
                                                            <mtd>
                                                                <mn>
                                                                    4
                                                                </mn>
                                                            </mtd>
                                                        </mtr>
                                                        <mtr>
                                                            <mtd>
                                                                <mn>
                                                                    2
                                                                </mn>
                                                                <mo>
                                                                    −<!-- − -->
                                                                </mo>
                                                                <mi>
                                                                    i
                                                                </mi>
                                                            </mtd>
                                                            <mtd>
                                                                <mn>
                                                                    3
                                                                </mn>
                                                            </mtd>
                                                            <mtd>
                                                                <mi>
                                                                    i
                                                                </mi>
                                                            </mtd>
                                                        </mtr>
                                                        <mtr>
                                                            <mtd>
                                                                <mn>
                                                                    4
                                                                </mn>
                                                            </mtd>
                                                            <mtd>
                                                                <mo>
                                                                    −<!-- − -->
                                                                </mo>
                                                                <mi>
                                                                    i
                                                                </mi>
                                                            </mtd>
                                                            <mtd>
                                                                <mn>
                                                                    1
                                                                </mn>
                                                            </mtd>
                                                        </mtr>
                                                    </mtable>
                                                    <mo>
                                                        ]
                                                    </mo>
                                                </mrow>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ccf11c16396b6ddd4f2254f7132cd8bb2c57ee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.005ex; width:19.27ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}" /></span>
                            </dd>
                        </dl>
                        <p>
                            The diagonal elements must be <a href="/wiki/Real_number" title="Real number">real</a>, as they must be their own complex conjugate.
                        </p>
                        <p>
                            Well-known families of <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, <a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a> and their generalizations are Hermitian. In <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> such Hermitian matrices are often multiplied by <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> coefficients,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup> which results in <i>skew-Hermitian</i> matrices (see <a href="#facts">below</a>).
                        </p>
                        <p>
                            Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> equals the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication of a matrix</a> and its conjugate transpose, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=BB^{\mathsf {H}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                        <mo>
                                            =
                                        </mo>
                                        <mi>
                                            B
                                        </mi>
                                        <msup>
                                            <mi>
                                                B
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        H
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A=BB^{\mathsf {H}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0efab2d7c3a4b4b7caf14cc0705dadd13195a9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.765ex; height:2.676ex;" alt="{\displaystyle A=BB^{\mathsf {H}}}" /></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is a Hermitian <a href="/wiki/Positive_semi-definite_matrix" class="mw-redirect" title="Positive semi-definite matrix">positive semi-definite matrix</a>. Furthermore, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            B
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle B}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="B" /></span> is row full-rank, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            A
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle A}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is positive definite.
                        </p>
                        <h2>
                            <span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=7" title="Edit section: Properties">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <table class="box-Expand_section plainlinks metadata ambox ambox-content" role="presentation">
                            <tbody>
                                <tr>
                                    <td class="mbox-image">
                                        <div style="width:52px">
                                            <a href="/wiki/File:Wiki_letter_w_cropped.svg" class="image"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/44px-Wiki_letter_w_cropped.svg.png" decoding="async" width="44" height="31" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/66px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/88px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a>
                                        </div>
                                    </td>
                                    <td class="mbox-text">
                                        <div class="mbox-text-span">
                                            This section <b>needs expansion</b> with: Proof of the properties requested. <small>You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=1">adding to it</a>.</small> <small class="date-container"><i>(<span class="date">February 2018</span>)</i></small>
                                        </div>
                                    </td>
                                </tr>
                            </tbody>
                        </table>
                        <ul>
                            <li>The entries on the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> (top left to bottom right) of any Hermitian matrix are <a href="/wiki/Real_number" title="Real number">real</a>.
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> By definition of the Hermitian matrix
                                <dl>
                                    <dd>
                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}={\overline {H}}_{ji}}">
                                        <semantics>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mstyle displaystyle="true" scriptlevel="0">
                                                    <msub>
                                                        <mi>
                                                            H
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                i
                                                            </mi>
                                                            <mi>
                                                                j
                                                            </mi>
                                                        </mrow>
                                                    </msub>
                                                    <mo>
                                                        =
                                                    </mo>
                                                    <msub>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mover>
                                                                <mi>
                                                                    H
                                                                </mi>
                                                                <mo accent="false">
                                                                    ¯<!-- ¯ -->
                                                                </mo>
                                                            </mover>
                                                        </mrow>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                            <mi>
                                                                i
                                                            </mi>
                                                        </mrow>
                                                    </msub>
                                                </mstyle>
                                            </mrow>
                                            <annotation encoding="application/x-tex">
                                                {\displaystyle H_{ij}={\overline {H}}_{ji}}
                                            </annotation>
                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}" /></span>
                                    </dd>
                                </dl>
                            </dd>
                            <dd>
                                so for <span class="texhtml"><i>i</i> = <i>j</i></span> the above follows.
                            </dd>
                            <dd>
                                Only the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their <a href="/wiki/Off-diagonal_element" class="mw-redirect" title="Off-diagonal element">off-diagonal elements</a>, as long as diagonally-opposite entries are complex conjugates.
                            </dd>
                        </dl>
                        <ul>
                            <li>A matrix that has only real entries is Hermitian <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>. A real and symmetric matrix is simply a special case of a Hermitian matrix.
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}={\overline {H}}_{ji}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msub>
                                                <mi>
                                                    H
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                =
                                            </mo>
                                            <msub>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mover>
                                                        <mi>
                                                            H
                                                        </mi>
                                                        <mo accent="false">
                                                            ¯<!-- ¯ -->
                                                        </mo>
                                                    </mover>
                                                </mrow>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mi>
                                                        i
                                                    </mi>
                                                </mrow>
                                            </msub>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle H_{ij}={\overline {H}}_{ji}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}" /></span> by definition. Thus <span class="texhtml">H<sub><i>ij</i></sub> = H<sub><i>ji</i></sub></span> (matrix symmetry) if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}={\overline {H}}_{ij}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msub>
                                                <mi>
                                                    H
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                =
                                            </mo>
                                            <msub>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mover>
                                                        <mi>
                                                            H
                                                        </mi>
                                                        <mo accent="false">
                                                            ¯<!-- ¯ -->
                                                        </mo>
                                                    </mover>
                                                </mrow>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle H_{ij}={\overline {H}}_{ij}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1862750b96d01100244370b3fca45f01923ce5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ij}}" /></span> (<span class="texhtml">H<sub><i>ij</i></sub></span> is real).
                            </dd>
                        </dl>
                        <ul>
                            <li>Every Hermitian matrix is a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>. That is to say, <span class="texhtml">AA<sup>H</sup> = A<sup>H</sup>A</span>.
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> <span class="texhtml">A = A<sup>H</sup></span>, so <span class="texhtml">AA<sup>H</sup> = AA = A<sup>H</sup>A</span>.
                            </dd>
                        </dl>
                        <ul>
                            <li>The finite-dimensional <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> says that any Hermitian matrix can be <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalized</a> by a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>, and that the resulting diagonal matrix has only real entries. This implies that all <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvalues</a> of a Hermitian matrix <span class="texhtml mvar" style="font-style:italic;">A</span> with dimension <span class="texhtml mvar" style="font-style:italic;">n</span> are real, and that <span class="texhtml mvar" style="font-style:italic;">A</span> has <span class="texhtml mvar" style="font-style:italic;">n</span> linearly independent <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a>. Moreover, a Hermitian matrix has <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a> of <span class="texhtml">ℂ<sup><i>n</i></sup></span> consisting of <span class="texhtml mvar" style="font-style:italic;">n</span> eigenvectors of <span class="texhtml mvar" style="font-style:italic;">A</span>.
                            </li>
                        </ul>
                        <ul>
                            <li>The sum of any two Hermitian matrices is Hermitian.
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                +
                                            </mo>
                                            <mi>
                                                B
                                            </mi>
                                            <msub>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                =
                                            </mo>
                                            <msub>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                +
                                            </mo>
                                            <msub>
                                                <mi>
                                                    B
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        i
                                                    </mi>
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                =
                                            </mo>
                                            <msub>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mover>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mo accent="false">
                                                            ¯<!-- ¯ -->
                                                        </mo>
                                                    </mover>
                                                </mrow>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mi>
                                                        i
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                +
                                            </mo>
                                            <msub>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mover>
                                                        <mi>
                                                            B
                                                        </mi>
                                                        <mo accent="false">
                                                            ¯<!-- ¯ -->
                                                        </mo>
                                                    </mover>
                                                </mrow>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mi>
                                                        i
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                =
                                            </mo>
                                            <msub>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mover>
                                                        <mrow>
                                                            <mo stretchy="false">
                                                                (
                                                            </mo>
                                                            <mi>
                                                                A
                                                            </mi>
                                                            <mo>
                                                                +
                                                            </mo>
                                                            <mi>
                                                                B
                                                            </mi>
                                                            <mo stretchy="false">
                                                                )
                                                            </mo>
                                                        </mrow>
                                                        <mo accent="false">
                                                            ¯<!-- ¯ -->
                                                        </mo>
                                                    </mover>
                                                </mrow>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mi>
                                                        i
                                                    </mi>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                ,
                                            </mo>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251bf4ebbe3b0d119e0a7d19f8fd134c4f072971" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:48.159ex; height:4.176ex;" alt="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}" /></span> as claimed.
                            </dd>
                        </dl>
                        <ul>
                            <li>The <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a> of an invertible Hermitian matrix is Hermitian as well.
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}A=I}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                I
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A^{-1}A=I}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021893240ff7fa3148b6649b0ba4d88cd207b5f0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.089ex; height:2.676ex;" alt="{\displaystyle A^{-1}A=I}" /></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                I
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mi>
                                                    I
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        H
                                                    </mi>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msup>
                                            <mi>
                                                A
                                            </mi>
                                            <msup>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        H
                                                    </mi>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        H
                                                    </mi>
                                                </mrow>
                                            </msup>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msup>
                                            <msup>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        H
                                                    </mi>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msup>
                                            <msup>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        H
                                                    </mi>
                                                </mrow>
                                            </msup>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28a8250ab35ad60228bb0376eb4b7024f027815" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:46.124ex; height:3.176ex;" alt="{\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}" /></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}=(A^{-1})^{H}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msup>
                                            <msup>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        H
                                                    </mi>
                                                </mrow>
                                            </msup>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A^{-1}=(A^{-1})^{H}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0179c3a7aebe194ccd9a19ba02b972500f88a7a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.751ex; height:3.176ex;" alt="{\displaystyle A^{-1}=(A^{-1})^{H}}" /></span> as claimed.
                            </dd>
                        </dl>
                        <ul>
                            <li>The <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">product</a> of two Hermitian matrices <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> is Hermitian if and only if <span class="texhtml"><i>AB</i> = <i>BA</i></span>.
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                B
                                            </mi>
                                            <msup>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <mrow>
                                                        <mo stretchy="false">
                                                            (
                                                        </mo>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mi>
                                                            B
                                                        </mi>
                                                        <msup>
                                                            <mo stretchy="false">
                                                                )
                                                            </mo>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi mathvariant="sans-serif">
                                                                        T
                                                                    </mi>
                                                                </mrow>
                                                            </mrow>
                                                        </msup>
                                                    </mrow>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <mrow>
                                                        <msup>
                                                            <mi>
                                                                B
                                                            </mi>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi mathvariant="sans-serif">
                                                                        T
                                                                    </mi>
                                                                </mrow>
                                                            </mrow>
                                                        </msup>
                                                        <msup>
                                                            <mi>
                                                                A
                                                            </mi>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi mathvariant="sans-serif">
                                                                        T
                                                                    </mi>
                                                                </mrow>
                                                            </mrow>
                                                        </msup>
                                                    </mrow>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <msup>
                                                        <mi>
                                                            B
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mi mathvariant="sans-serif">
                                                                    T
                                                                </mi>
                                                            </mrow>
                                                        </mrow>
                                                    </msup>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <msup>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mi mathvariant="sans-serif">
                                                                    T
                                                                </mi>
                                                            </mrow>
                                                        </mrow>
                                                    </msup>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mi>
                                                    B
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                B
                                            </mi>
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                .
                                            </mo>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cf8185ca7a0687bf959bb65b16db6cf1714ca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:52.205ex; height:4.009ex;" alt="{\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}" /></span> Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (AB)^{\mathsf {H}}=AB}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                B
                                            </mi>
                                            <msup>
                                                <mo stretchy="false">
                                                    )
                                                </mo>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                B
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle (AB)^{\mathsf {H}}=AB}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d303a1ebcac8547489b170be5d0dd7d8e04e548e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:3.176ex;" alt="{\displaystyle (AB)^{\mathsf {H}}=AB}" /></span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB=BA}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                B
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                B
                                            </mi>
                                            <mi>
                                                A
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle AB=BA}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992c8ea49fdd26b491180036c5a4d879fec77442" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.113ex; height:2.176ex;" alt="AB=BA" /></span>.
                            </dd>
                            <dd>
                                Thus <span class="texhtml"><i>A</i><sup><i>n</i></sup></span> is Hermitian if <span class="texhtml mvar" style="font-style:italic;">A</span> is Hermitian and <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer.
                            </dd>
                        </dl>
                        <ul>
                            <li>For an arbitrary complex valued vector <span class="texhtml mvar" style="font-style:italic;">v</span> the product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{\mathsf {H}}Av}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msup>
                                                <mi>
                                                    v
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                v
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle v^{\mathsf {H}}Av}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c23d6757968a2267ee906cffc07cfe1bbc8aecc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.394ex; height:2.676ex;" alt="{\displaystyle v^{\mathsf {H}}Av}" /></span> is real because of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{\mathsf {H}}Av=\left(v^{\mathsf {H}}Av\right)^{\mathsf {H}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msup>
                                                <mi>
                                                    v
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mi>
                                                A
                                            </mi>
                                            <mi>
                                                v
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mrow>
                                                    <mo>
                                                        (
                                                    </mo>
                                                    <mrow>
                                                        <msup>
                                                            <mi>
                                                                v
                                                            </mi>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi mathvariant="sans-serif">
                                                                        H
                                                                    </mi>
                                                                </mrow>
                                                            </mrow>
                                                        </msup>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mi>
                                                            v
                                                        </mi>
                                                    </mrow>
                                                    <mo>
                                                        )
                                                    </mo>
                                                </mrow>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle v^{\mathsf {H}}Av=\left(v^{\mathsf {H}}Av\right)^{\mathsf {H}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7839ef3cdbc89c3ea1acd17a507d03a33ed79df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:17.412ex; height:3.843ex;" alt="{\displaystyle v^{\mathsf {H}}Av=\left(v^{\mathsf {H}}Av\right)^{\mathsf {H}}}" /></span>. This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system e.g. total <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> which have to be real.
                            </li>
                        </ul>
                        <ul>
                            <li>The Hermitian complex <span class="texhtml mvar" style="font-style:italic;">n</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> matrices do not form a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, <span class="texhtml">ℂ</span>, since the identity matrix <span class="texhtml"><i>I</i><sub><i>n</i></sub></span> is Hermitian, but <span class="texhtml"><i>i</i> <i>I</i><sub><i>n</i></sub></span> is not. However the complex Hermitian matrices <i>do</i> form a vector space over the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> <span class="texhtml">ℝ</span>. In the <span class="texhtml">2<i>n</i><sup>2</sup></span>-<a href="/wiki/Dimension_of_a_vector_space" class="mw-redirect" title="Dimension of a vector space">dimensional</a> vector space of complex <span class="texhtml"><i>n</i> × <i>n</i></span> matrices over <span class="texhtml">ℝ</span>, the complex Hermitian matrices form a subspace of dimension <span class="texhtml"><i>n</i><sup>2</sup></span>. If <span class="texhtml"><i>E</i><sub><i>jk</i></sub></span> denotes the <span class="texhtml mvar" style="font-style:italic;">n</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> matrix with a <span class="texhtml">1</span> in the <span class="texhtml"><i>j</i>,<i>k</i></span> position and zeros elsewhere, a basis (orthonormal w.r.t. the Frobenius inner product) can be described as follows:
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <dl>
                                    <dd>
                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}">
                                        <semantics>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mstyle displaystyle="true" scriptlevel="0">
                                                    <msub>
                                                        <mi>
                                                            E
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                            <mi>
                                                                j
                                                            </mi>
                                                        </mrow>
                                                    </msub>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mtext>
                                                            &nbsp;for&nbsp;
                                                        </mtext>
                                                    </mrow>
                                                    <mn>
                                                        1
                                                    </mn>
                                                    <mo>
                                                        ≤<!-- ≤ -->
                                                    </mo>
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mo>
                                                        ≤<!-- ≤ -->
                                                    </mo>
                                                    <mi>
                                                        n
                                                    </mi>
                                                    <mspace width="1em"></mspace>
                                                    <mo stretchy="false">
                                                        (
                                                    </mo>
                                                    <mi>
                                                        n
                                                    </mi>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mtext>
                                                            &nbsp;matrices
                                                        </mtext>
                                                    </mrow>
                                                    <mo stretchy="false">
                                                        )
                                                    </mo>
                                                </mstyle>
                                            </mrow>
                                            <annotation encoding="application/x-tex">
                                                {\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}
                                            </annotation>
                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eedb181c0bdae46e8c1526161b03d0ea97b4b4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:31.612ex; height:3.009ex;" alt="{\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}" /></span>
                                    </dd>
                                </dl>
                            </dd>
                        </dl>
                        <dl>
                            <dd>
                                together with the set of matrices of the form
                            </dd>
                        </dl>
                        <dl>
                            <dd>
                                <dl>
                                    <dd>
                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}">
                                        <semantics>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mstyle displaystyle="true" scriptlevel="0">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mfrac>
                                                            <mn>
                                                                1
                                                            </mn>
                                                            <msqrt>
                                                                <mn>
                                                                    2
                                                                </mn>
                                                            </msqrt>
                                                        </mfrac>
                                                    </mrow>
                                                    <mrow>
                                                        <mo>
                                                            (
                                                        </mo>
                                                        <mrow>
                                                            <msub>
                                                                <mi>
                                                                    E
                                                                </mi>
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi>
                                                                        j
                                                                    </mi>
                                                                    <mi>
                                                                        k
                                                                    </mi>
                                                                </mrow>
                                                            </msub>
                                                            <mo>
                                                                +
                                                            </mo>
                                                            <msub>
                                                                <mi>
                                                                    E
                                                                </mi>
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi>
                                                                        k
                                                                    </mi>
                                                                    <mi>
                                                                        j
                                                                    </mi>
                                                                </mrow>
                                                            </msub>
                                                        </mrow>
                                                        <mo>
                                                            )
                                                        </mo>
                                                    </mrow>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mtext>
                                                            &nbsp;for&nbsp;
                                                        </mtext>
                                                    </mrow>
                                                    <mn>
                                                        1
                                                    </mn>
                                                    <mo>
                                                        ≤<!-- ≤ -->
                                                    </mo>
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mo>
                                                        &lt;
                                                    </mo>
                                                    <mi>
                                                        k
                                                    </mi>
                                                    <mo>
                                                        ≤<!-- ≤ -->
                                                    </mo>
                                                    <mi>
                                                        n
                                                    </mi>
                                                    <mspace width="1em"></mspace>
                                                    <mrow>
                                                        <mo>
                                                            (
                                                        </mo>
                                                        <mrow>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mfrac>
                                                                    <mrow>
                                                                        <msup>
                                                                            <mi>
                                                                                n
                                                                            </mi>
                                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                                <mn>
                                                                                    2
                                                                                </mn>
                                                                            </mrow>
                                                                        </msup>
                                                                        <mo>
                                                                            −<!-- − -->
                                                                        </mo>
                                                                        <mi>
                                                                            n
                                                                        </mi>
                                                                    </mrow>
                                                                    <mn>
                                                                        2
                                                                    </mn>
                                                                </mfrac>
                                                            </mrow>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mtext>
                                                                    &nbsp;matrices
                                                                </mtext>
                                                            </mrow>
                                                        </mrow>
                                                        <mo>
                                                            )
                                                        </mo>
                                                    </mrow>
                                                </mstyle>
                                            </mrow>
                                            <annotation encoding="application/x-tex">
                                                {\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}
                                            </annotation>
                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeac51c423f6dbefc5f63e483d9aee96e6fa342" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:57.064ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}" /></span>
                                    </dd>
                                </dl>
                            </dd>
                        </dl>
                        <dl>
                            <dd>
                                and the matrices
                            </dd>
                        </dl>
                        <dl>
                            <dd>
                                <dl>
                                    <dd>
                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}">
                                        <semantics>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mstyle displaystyle="true" scriptlevel="0">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mfrac>
                                                            <mi>
                                                                i
                                                            </mi>
                                                            <msqrt>
                                                                <mn>
                                                                    2
                                                                </mn>
                                                            </msqrt>
                                                        </mfrac>
                                                    </mrow>
                                                    <mrow>
                                                        <mo>
                                                            (
                                                        </mo>
                                                        <mrow>
                                                            <msub>
                                                                <mi>
                                                                    E
                                                                </mi>
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi>
                                                                        j
                                                                    </mi>
                                                                    <mi>
                                                                        k
                                                                    </mi>
                                                                </mrow>
                                                            </msub>
                                                            <mo>
                                                                −<!-- − -->
                                                            </mo>
                                                            <msub>
                                                                <mi>
                                                                    E
                                                                </mi>
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi>
                                                                        k
                                                                    </mi>
                                                                    <mi>
                                                                        j
                                                                    </mi>
                                                                </mrow>
                                                            </msub>
                                                        </mrow>
                                                        <mo>
                                                            )
                                                        </mo>
                                                    </mrow>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mtext>
                                                            &nbsp;for&nbsp;
                                                        </mtext>
                                                    </mrow>
                                                    <mn>
                                                        1
                                                    </mn>
                                                    <mo>
                                                        ≤<!-- ≤ -->
                                                    </mo>
                                                    <mi>
                                                        j
                                                    </mi>
                                                    <mo>
                                                        &lt;
                                                    </mo>
                                                    <mi>
                                                        k
                                                    </mi>
                                                    <mo>
                                                        ≤<!-- ≤ -->
                                                    </mo>
                                                    <mi>
                                                        n
                                                    </mi>
                                                    <mspace width="1em"></mspace>
                                                    <mrow>
                                                        <mo>
                                                            (
                                                        </mo>
                                                        <mrow>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mfrac>
                                                                    <mrow>
                                                                        <msup>
                                                                            <mi>
                                                                                n
                                                                            </mi>
                                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                                <mn>
                                                                                    2
                                                                                </mn>
                                                                            </mrow>
                                                                        </msup>
                                                                        <mo>
                                                                            −<!-- − -->
                                                                        </mo>
                                                                        <mi>
                                                                            n
                                                                        </mi>
                                                                    </mrow>
                                                                    <mn>
                                                                        2
                                                                    </mn>
                                                                </mfrac>
                                                            </mrow>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mtext>
                                                                    &nbsp;matrices
                                                                </mtext>
                                                            </mrow>
                                                        </mrow>
                                                        <mo>
                                                            )
                                                        </mo>
                                                    </mrow>
                                                </mstyle>
                                            </mrow>
                                            <annotation encoding="application/x-tex">
                                                {\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}
                                            </annotation>
                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db65cce3a8fa33e5b7b96badd756c8573aa866c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:57.064ex; height:6.676ex;" alt="{\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}" /></span>
                                    </dd>
                                </dl>
                            </dd>
                        </dl>
                        <dl>
                            <dd>
                                where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                i
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle i}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="i" /></span> denotes the complex number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <msqrt>
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <mn>
                                                        1
                                                    </mn>
                                                </msqrt>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle {\sqrt {-1}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\sqrt {-1}}" /></span>, called the <i><a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a></i>.
                            </dd>
                        </dl>
                        <ul>
                            <li>If <span class="texhtml mvar" style="font-style:italic;">n</span> orthonormal eigenvectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1},\dots ,u_{n}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msub>
                                                <mi>
                                                    u
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mn>
                                                        1
                                                    </mn>
                                                </mrow>
                                            </msub>
                                            <mo>
                                                ,
                                            </mo>
                                            <mo>
                                                …<!-- … -->
                                            </mo>
                                            <mo>
                                                ,
                                            </mo>
                                            <msub>
                                                <mi>
                                                    u
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        n
                                                    </mi>
                                                </mrow>
                                            </msub>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle u_{1},\dots ,u_{n}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba291b64c0d3afa90b4556a7a601116dfd74ef2e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle u_{1},\dots ,u_{n}}" /></span> of a Hermitian matrix are chosen and written as the columns of the matrix <span class="texhtml mvar" style="font-style:italic;">U</span>, then one <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">eigendecomposition</a> of <span class="texhtml mvar" style="font-style:italic;">A</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=U\Lambda U^{\mathsf {H}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                U
                                            </mi>
                                            <mi mathvariant="normal">
                                                Λ<!-- Λ -->
                                            </mi>
                                            <msup>
                                                <mi>
                                                    U
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A=U\Lambda U^{\mathsf {H}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ef5bade4fb04b1aaf53c0d73e4763d4d154eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:11.474ex; height:2.676ex;" alt="{\displaystyle A=U\Lambda U^{\mathsf {H}}}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                U
                                            </mi>
                                            <msup>
                                                <mi>
                                                    U
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mo>
                                                =
                                            </mo>
                                            <mi>
                                                I
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mi>
                                                    U
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mi>
                                                U
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b819b015c1984e6cb07153c675815d4657b90da8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:17.408ex; height:2.676ex;" alt="{\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}" /></span> and therefore
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <dl>
                                    <dd>
                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}">
                                        <semantics>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mstyle displaystyle="true" scriptlevel="0">
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mo>
                                                        =
                                                    </mo>
                                                    <munder>
                                                        <mo>
                                                            ∑<!-- ∑ -->
                                                        </mo>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                        </mrow>
                                                    </munder>
                                                    <msub>
                                                        <mi>
                                                            λ<!-- λ -->
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                        </mrow>
                                                    </msub>
                                                    <msub>
                                                        <mi>
                                                            u
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                        </mrow>
                                                    </msub>
                                                    <msubsup>
                                                        <mi>
                                                            u
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi>
                                                                j
                                                            </mi>
                                                        </mrow>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mi mathvariant="sans-serif">
                                                                    H
                                                                </mi>
                                                            </mrow>
                                                        </mrow>
                                                    </msubsup>
                                                    <mo>
                                                        ,
                                                    </mo>
                                                </mstyle>
                                            </mrow>
                                            <annotation encoding="application/x-tex">
                                                {\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}
                                            </annotation>
                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7d749931e5f709bcbc0a446638d3b6b8ed0c6c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:16.46ex; height:5.843ex;" alt="{\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}" /></span>
                                    </dd>
                                </dl>
                            </dd>
                            <dd>
                                where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{j}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <msub>
                                                <mi>
                                                    λ<!-- λ -->
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi>
                                                        j
                                                    </mi>
                                                </mrow>
                                            </msub>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \lambda _{j}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa91daf9145f27bb95746fd2a37537342d587b77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.265ex; height:2.843ex;" alt="\lambda _{j}" /></span> are the eigenvalues on the diagonal of the diagonal matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\Lambda }">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mspace width="thickmathspace"></mspace>
                                            <mi mathvariant="normal">
                                                Λ<!-- Λ -->
                                            </mi>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \;\Lambda }
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1934e7eadd31fbf6f7d6bcf9c0e9bec934ce8976" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.258ex; height:2.176ex;" alt="\; \Lambda " /></span>.
                            </dd>
                        </dl>
                        <ul>
                            <li>The determinant of a Hermitian matrix is real:
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <i>Proof:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mo movablelimits="true" form="prefix">
                                                det
                                            </mo>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mo stretchy="false">
                                                )
                                            </mo>
                                            <mo>
                                                =
                                            </mo>
                                            <mo movablelimits="true" form="prefix">
                                                det
                                            </mo>
                                            <mrow>
                                                <mo>
                                                    (
                                                </mo>
                                                <msup>
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi mathvariant="sans-serif">
                                                                T
                                                            </mi>
                                                        </mrow>
                                                    </mrow>
                                                </msup>
                                                <mo>
                                                    )
                                                </mo>
                                            </mrow>
                                            <mspace width="1em"></mspace>
                                            <mo stretchy="false">
                                                ⇒<!-- ⇒ -->
                                            </mo>
                                            <mspace width="1em"></mspace>
                                            <mo movablelimits="true" form="prefix">
                                                det
                                            </mo>
                                            <mrow>
                                                <mo>
                                                    (
                                                </mo>
                                                <msup>
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mi mathvariant="sans-serif">
                                                                H
                                                            </mi>
                                                        </mrow>
                                                    </mrow>
                                                </msup>
                                                <mo>
                                                    )
                                                </mo>
                                            </mrow>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <mrow>
                                                        <mo movablelimits="true" form="prefix">
                                                            det
                                                        </mo>
                                                        <mo stretchy="false">
                                                            (
                                                        </mo>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mo stretchy="false">
                                                            )
                                                        </mo>
                                                    </mrow>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1240df64c3010e0be6eae865fdfcfe6f77bf5eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:45.862ex; height:3.843ex;" alt="{\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}" /></span>
                            </dd>
                            <dd>
                                Therefore if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                A
                                            </mi>
                                            <mo>
                                                =
                                            </mo>
                                            <msup>
                                                <mi>
                                                    A
                                                </mi>
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mi mathvariant="sans-serif">
                                                            H
                                                        </mi>
                                                    </mrow>
                                                </mrow>
                                            </msup>
                                            <mspace width="1em"></mspace>
                                            <mo stretchy="false">
                                                ⇒<!-- ⇒ -->
                                            </mo>
                                            <mspace width="1em"></mspace>
                                            <mo movablelimits="true" form="prefix">
                                                det
                                            </mo>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <mi>
                                                A
                                            </mi>
                                            <mo stretchy="false">
                                                )
                                            </mo>
                                            <mo>
                                                =
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mover>
                                                    <mrow>
                                                        <mo movablelimits="true" form="prefix">
                                                            det
                                                        </mo>
                                                        <mo stretchy="false">
                                                            (
                                                        </mo>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mo stretchy="false">
                                                            )
                                                        </mo>
                                                    </mrow>
                                                    <mo accent="false">
                                                        ¯<!-- ¯ -->
                                                    </mo>
                                                </mover>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cc392bdcfbb134dd66d9b469847f6370e29d9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:33.017ex; height:3.676ex;" alt="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}" /></span>.
                            </dd>
                            <dd>
                                (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
                            </dd>
                        </dl>
                        <h2>
                            <span class="mw-headline" id="Decomposition_into_Hermitian_and_skew-Hermitian">Decomposition into Hermitian and skew-Hermitian</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=8" title="Edit section: Decomposition into Hermitian and skew-Hermitian">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <p>
                            <span id="facts"></span>Additional facts related to Hermitian matrices include:
                        </p>
                        <ul>
                            <li>The sum of a square matrix and its conjugate transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A+A^{\mathsf {H}}\right)}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mrow>
                                                <mo>
                                                    (
                                                </mo>
                                                <mrow>
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mo>
                                                        +
                                                    </mo>
                                                    <msup>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mi mathvariant="sans-serif">
                                                                    H
                                                                </mi>
                                                            </mrow>
                                                        </mrow>
                                                    </msup>
                                                </mrow>
                                                <mo>
                                                    )
                                                </mo>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \left(A+A^{\mathsf {H}}\right)}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef97bb04ce4ab682bcc84cf1059f8da235b483e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:9.852ex; height:3.343ex;" alt="{\displaystyle \left(A+A^{\mathsf {H}}\right)}" /></span> is Hermitian.
                            </li>
                        </ul>
                        <ul>
                            <li>The difference of a square matrix and its conjugate transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A-A^{\mathsf {H}}\right)}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mrow>
                                                <mo>
                                                    (
                                                </mo>
                                                <mrow>
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mo>
                                                        −<!-- − -->
                                                    </mo>
                                                    <msup>
                                                        <mi>
                                                            A
                                                        </mi>
                                                        <mrow class="MJX-TeXAtom-ORD">
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mi mathvariant="sans-serif">
                                                                    H
                                                                </mi>
                                                            </mrow>
                                                        </mrow>
                                                    </msup>
                                                </mrow>
                                                <mo>
                                                    )
                                                </mo>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle \left(A-A^{\mathsf {H}}\right)}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ac665be4943ce769e33109e9f64abcf1e98050" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:9.852ex; height:3.343ex;" alt="{\displaystyle \left(A-A^{\mathsf {H}}\right)}" /></span> is <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">skew-Hermitian</a> (also called antihermitian). This implies that the <a href="/wiki/Commutator" title="Commutator">commutator</a> of two Hermitian matrices is skew-Hermitian.
                            </li>
                        </ul>
                        <ul>
                            <li>An arbitrary square matrix <span class="texhtml mvar" style="font-style:italic;">C</span> can be written as the sum of a Hermitian matrix <span class="texhtml mvar" style="font-style:italic;">A</span> and a skew-Hermitian matrix <span class="texhtml mvar" style="font-style:italic;">B</span>. This is known as the Toeplitz decomposition of <span class="texhtml mvar" style="font-style:italic;">C</span>.<sup id="cite_ref-HornJohnson_3-0" class="reference"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup class="reference" style="white-space:nowrap;">:<span>p. 7</span></sup>
                            </li>
                        </ul>
                        <dl>
                            <dd>
                                <dl>
                                    <dd>
                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}">
                                        <semantics>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mstyle displaystyle="true" scriptlevel="0">
                                                    <mi>
                                                        C
                                                    </mi>
                                                    <mo>
                                                        =
                                                    </mo>
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mo>
                                                        +
                                                    </mo>
                                                    <mi>
                                                        B
                                                    </mi>
                                                    <mspace width="1em"></mspace>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mstyle displaystyle="false" scriptlevel="0">
                                                            <mtext>
                                                                with
                                                            </mtext>
                                                        </mstyle>
                                                    </mrow>
                                                    <mspace width="1em"></mspace>
                                                    <mi>
                                                        A
                                                    </mi>
                                                    <mo>
                                                        =
                                                    </mo>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mfrac>
                                                            <mn>
                                                                1
                                                            </mn>
                                                            <mn>
                                                                2
                                                            </mn>
                                                        </mfrac>
                                                    </mrow>
                                                    <mrow>
                                                        <mo>
                                                            (
                                                        </mo>
                                                        <mrow>
                                                            <mi>
                                                                C
                                                            </mi>
                                                            <mo>
                                                                +
                                                            </mo>
                                                            <msup>
                                                                <mi>
                                                                    C
                                                                </mi>
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mrow class="MJX-TeXAtom-ORD">
                                                                        <mi mathvariant="sans-serif">
                                                                            H
                                                                        </mi>
                                                                    </mrow>
                                                                </mrow>
                                                            </msup>
                                                        </mrow>
                                                        <mo>
                                                            )
                                                        </mo>
                                                    </mrow>
                                                    <mspace width="1em"></mspace>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mstyle displaystyle="false" scriptlevel="0">
                                                            <mtext>
                                                                and
                                                            </mtext>
                                                        </mstyle>
                                                    </mrow>
                                                    <mspace width="1em"></mspace>
                                                    <mi>
                                                        B
                                                    </mi>
                                                    <mo>
                                                        =
                                                    </mo>
                                                    <mrow class="MJX-TeXAtom-ORD">
                                                        <mfrac>
                                                            <mn>
                                                                1
                                                            </mn>
                                                            <mn>
                                                                2
                                                            </mn>
                                                        </mfrac>
                                                    </mrow>
                                                    <mrow>
                                                        <mo>
                                                            (
                                                        </mo>
                                                        <mrow>
                                                            <mi>
                                                                C
                                                            </mi>
                                                            <mo>
                                                                −<!-- − -->
                                                            </mo>
                                                            <msup>
                                                                <mi>
                                                                    C
                                                                </mi>
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mrow class="MJX-TeXAtom-ORD">
                                                                        <mi mathvariant="sans-serif">
                                                                            H
                                                                        </mi>
                                                                    </mrow>
                                                                </mrow>
                                                            </msup>
                                                        </mrow>
                                                        <mo>
                                                            )
                                                        </mo>
                                                    </mrow>
                                                </mstyle>
                                            </mrow>
                                            <annotation encoding="application/x-tex">
                                                {\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}
                                            </annotation>
                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0919d2e50fe1008af261f8301f243c002c328dbf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:63.495ex; height:5.176ex;" alt="{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}" /></span>
                                    </dd>
                                </dl>
                            </dd>
                        </dl>
                        <h2>
                            <span class="mw-headline" id="Rayleigh_quotient">Rayleigh quotient</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=9" title="Edit section: Rayleigh quotient">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <div role="note" class="hatnote navigation-not-searchable">
                            Main article: <a href="/wiki/Rayleigh_quotient" title="Rayleigh quotient">Rayleigh quotient</a>
                        </div>
                        <p>
                            In mathematics, for a given complex Hermitian matrix <i>M</i> and nonzero vector <i>x</i>, the Rayleigh quotient<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,x)}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            R
                                        </mi>
                                        <mo stretchy="false">
                                            (
                                        </mo>
                                        <mi>
                                            M
                                        </mi>
                                        <mo>
                                            ,
                                        </mo>
                                        <mi>
                                            x
                                        </mi>
                                        <mo stretchy="false">
                                            )
                                        </mo>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle R(M,x)}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ed067bb4bc06662d6bdf6210d450779a529ce5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.379ex; height:2.843ex;" alt="R(M, x)" /></span>, is defined as:<sup id="cite_ref-HornJohnson_3-1" class="reference"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup class="reference" style="white-space:nowrap;">:<span>p. 234</span></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup>
                        </p>
                        <dl>
                            <dd>
                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}">
                                <semantics>
                                    <mrow class="MJX-TeXAtom-ORD">
                                        <mstyle displaystyle="true" scriptlevel="0">
                                            <mi>
                                                R
                                            </mi>
                                            <mo stretchy="false">
                                                (
                                            </mo>
                                            <mi>
                                                M
                                            </mi>
                                            <mo>
                                                ,
                                            </mo>
                                            <mi>
                                                x
                                            </mi>
                                            <mo stretchy="false">
                                                )
                                            </mo>
                                            <mo>
                                                :=
                                            </mo>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mfrac>
                                                    <mrow>
                                                        <msup>
                                                            <mi>
                                                                x
                                                            </mi>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi mathvariant="sans-serif">
                                                                        H
                                                                    </mi>
                                                                </mrow>
                                                            </mrow>
                                                        </msup>
                                                        <mi>
                                                            M
                                                        </mi>
                                                        <mi>
                                                            x
                                                        </mi>
                                                    </mrow>
                                                    <mrow>
                                                        <msup>
                                                            <mi>
                                                                x
                                                            </mi>
                                                            <mrow class="MJX-TeXAtom-ORD">
                                                                <mrow class="MJX-TeXAtom-ORD">
                                                                    <mi mathvariant="sans-serif">
                                                                        H
                                                                    </mi>
                                                                </mrow>
                                                            </mrow>
                                                        </msup>
                                                        <mi>
                                                            x
                                                        </mi>
                                                    </mrow>
                                                </mfrac>
                                            </mrow>
                                        </mstyle>
                                    </mrow>
                                    <annotation encoding="application/x-tex">
                                        {\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}
                                    </annotation>
                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9b0047f8437f7b012041d7b2fcd190a5a9ec2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:19.458ex; height:6.009ex;" alt="{\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}" /></span>.
                            </dd>
                        </dl>
                        <p>
                            For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\mathsf {H}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                x
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        H
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle x^{\mathsf {H}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b431248ab2f121914608bbd1c2376715cecda9c8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.726ex; height:2.676ex;" alt="{\displaystyle x^{\mathsf {H}}}" /></span> to the usual transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\mathsf {T}}}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msup>
                                            <mi>
                                                x
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mrow class="MJX-TeXAtom-ORD">
                                                    <mi mathvariant="sans-serif">
                                                        T
                                                    </mi>
                                                </mrow>
                                            </mrow>
                                        </msup>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle x^{\mathsf {T}}}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ee4832d06e8560510d81237d0650c897d476e9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.681ex; height:2.676ex;" alt="{\displaystyle x^{\mathsf {T}}}" /></span>. Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,cx)=R(M,x)}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            R
                                        </mi>
                                        <mo stretchy="false">
                                            (
                                        </mo>
                                        <mi>
                                            M
                                        </mi>
                                        <mo>
                                            ,
                                        </mo>
                                        <mi>
                                            c
                                        </mi>
                                        <mi>
                                            x
                                        </mi>
                                        <mo stretchy="false">
                                            )
                                        </mo>
                                        <mo>
                                            =
                                        </mo>
                                        <mi>
                                            R
                                        </mi>
                                        <mo stretchy="false">
                                            (
                                        </mo>
                                        <mi>
                                            M
                                        </mi>
                                        <mo>
                                            ,
                                        </mo>
                                        <mi>
                                            x
                                        </mi>
                                        <mo stretchy="false">
                                            )
                                        </mo>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle R(M,cx)=R(M,x)}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d54d3c850d35f99329591e3b57cef98d17237f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.864ex; height:2.843ex;" alt="{\displaystyle R(M,cx)=R(M,x)}" /></span> for any non-zero real scalar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            c
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle c}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="c" /></span>. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
                        </p>
                        <p>
                            It can be shown<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2019)">citation needed</span></a></i>]</sup> that, for a given matrix, the Rayleigh quotient reaches its minimum value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\min }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msub>
                                            <mi>
                                                λ<!-- λ -->
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo movablelimits="true" form="prefix">
                                                    min
                                                </mo>
                                            </mrow>
                                        </msub>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle \lambda _{\min }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c24522483ceaf1d54224b69af4244b60c3ac08" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.328ex; height:2.509ex;" alt="\lambda_\min" /></span> (the smallest eigenvalue of M) when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            x
                                        </mi>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle x}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x" /></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\min }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msub>
                                            <mi>
                                                v
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo movablelimits="true" form="prefix">
                                                    min
                                                </mo>
                                            </mrow>
                                        </msub>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle v_{\min }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486623019ef451e0582b874018e0249a46e0f996" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.1ex; height:2.009ex;" alt="v_\min" /></span> (the corresponding eigenvector). Similarly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,x)\leq \lambda _{\max }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            R
                                        </mi>
                                        <mo stretchy="false">
                                            (
                                        </mo>
                                        <mi>
                                            M
                                        </mi>
                                        <mo>
                                            ,
                                        </mo>
                                        <mi>
                                            x
                                        </mi>
                                        <mo stretchy="false">
                                            )
                                        </mo>
                                        <mo>
                                            ≤<!-- ≤ -->
                                        </mo>
                                        <msub>
                                            <mi>
                                                λ<!-- λ -->
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo movablelimits="true" form="prefix">
                                                    max
                                                </mo>
                                            </mrow>
                                        </msub>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle R(M,x)\leq \lambda _{\max }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fbf88c578fc9f75d4610ebd18ab55f4f2842ce" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.124ex; height:2.843ex;" alt="R(M, x) \leq \lambda_\max" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,v_{\max })=\lambda _{\max }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <mi>
                                            R
                                        </mi>
                                        <mo stretchy="false">
                                            (
                                        </mo>
                                        <mi>
                                            M
                                        </mi>
                                        <mo>
                                            ,
                                        </mo>
                                        <msub>
                                            <mi>
                                                v
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo movablelimits="true" form="prefix">
                                                    max
                                                </mo>
                                            </mrow>
                                        </msub>
                                        <mo stretchy="false">
                                            )
                                        </mo>
                                        <mo>
                                            =
                                        </mo>
                                        <msub>
                                            <mi>
                                                λ<!-- λ -->
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo movablelimits="true" form="prefix">
                                                    max
                                                </mo>
                                            </mrow>
                                        </msub>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle R(M,v_{\max })=\lambda _{\max }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200db82bfdbc81cd227cb3470aa826d6f11a7653" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.213ex; height:2.843ex;" alt="R(M, v_\max) = \lambda_\max" /></span>.
                        </p>
                        <p>
                            The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
                        </p>
                        <p>
                            The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\max }}">
                            <semantics>
                                <mrow class="MJX-TeXAtom-ORD">
                                    <mstyle displaystyle="true" scriptlevel="0">
                                        <msub>
                                            <mi>
                                                λ<!-- λ -->
                                            </mi>
                                            <mrow class="MJX-TeXAtom-ORD">
                                                <mo movablelimits="true" form="prefix">
                                                    max
                                                </mo>
                                            </mrow>
                                        </msub>
                                    </mstyle>
                                </mrow>
                                <annotation encoding="application/x-tex">
                                    {\displaystyle \lambda _{\max }}
                                </annotation>
                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957584ae6a35f9edf293cb486d7436fb5b75e803" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.646ex; height:2.509ex;" alt="\lambda_\max" /></span> is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to <span class="texhtml"><i>M</i></span> associates the Rayleigh quotient <span class="texhtml"><i>R</i>(<i>M</i>, <i>x</i>)</span> for a fixed <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>M</i></span> varying through the algebra would be referred to as "vector state" of the algebra.
                        </p>
                        <h2>
                            <span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=10" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <ul>
                            <li>
                                <a href="/wiki/Vector_space" title="Vector space">Vector space</a>
                            </li>
                            <li>
                                <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian matrix</a> (anti-Hermitian matrix)
                            </li>
                            <li>
                                <a href="/wiki/Haynsworth_inertia_additivity_formula" title="Haynsworth inertia additivity formula">Haynsworth inertia additivity formula</a>
                            </li>
                            <li>
                                <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a>
                            </li>
                            <li>
                                <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint operator</a>
                            </li>
                            <li>
                                <a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary matrix</a>
                            </li>
                        </ul>
                        <h2>
                            <span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=11" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <div class="reflist" style="list-style-type: decimal;">
                            <div class="mw-references-wrap">
                                <ol class="references">
                                    <li id="cite_note-1">
                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Theodore_Frankel" title="Theodore Frankel">Frankel, Theodore</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DUnjs6nEn8wC&amp;lpg=PA652&amp;dq=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&amp;pg=PA652#v=onepage&amp;q&amp;f=false"><i>The Geometry of Physics: an introduction</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p.&#160;652. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-521-53927-7" title="Special:BookSources/0-521-53927-7"><bdi>0-521-53927-7</bdi></a>.</cite></span>
                                    </li>
                                    <li id="cite_note-2">
                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf">Physics 125 Course Notes</a> at <a href="/wiki/California_Institute_of_Technology" title="California Institute of Technology">California Institute of Technology</a></span>
                                    </li>
                                    <li id="cite_note-HornJohnson-3">
                                        <span class="mw-cite-backlink">^ <a href="#cite_ref-HornJohnson_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HornJohnson_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">Horn, Roger A.; Johnson, Charles R. (2013). <i>Matrix Analysis, second edition</i>. Cambridge University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780521839402" title="Special:BookSources/9780521839402"><bdi>9780521839402</bdi></a>.</cite></span>
                                        <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r935243608" />
                                    </li>
                                    <li id="cite_note-4">
                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Also known as the <b>Rayleigh–Ritz ratio</b>; named after <a href="/wiki/Walther_Ritz" title="Walther Ritz">Walther Ritz</a> and <a href="/wiki/Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a>.</span>
                                    </li>
                                    <li id="cite_note-5">
                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Parlet B. N. <i>The symmetric eigenvalue problem</i>, SIAM, Classics in Applied Mathematics,1998</span>
                                    </li>
                                </ol>
                            </div>
                        </div>
                        <h2>
                            <span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=12" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span>
                        </h2>
                        <ul>
                            <li>
                                <cite id="CITEREFHazewinkel2001" class="citation"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, ed. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=p/h047070">"Hermitian matrix"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer Science+Business Media B.V. / Kluwer Academic Publishers, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-1-55608-010-4" title="Special:BookSources/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a></cite>
                                <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r935243608" />
                            </li>
                            <li>
                                <a rel="nofollow" class="external text" href="https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Visualizing Hermitian Matrix as An Ellipse with Dr. Geo</a>, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
                            </li>
                            <li>
                                <cite class="citation web"><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath306/kmath306.htm">"Hermitian Matrices"</a>. <i>MathPages.com</i>.</cite>
                                <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r935243608" />
                            </li>
                        </ul><!-- 
NewPP limit report
Parsed by mw1375
Cached time: 20200224203345
Cache expiry: 2592000
Dynamic content: false
Complications: [vary‐revision‐sha1]
CPU time usage: 0.344 seconds
Real time usage: 0.623 seconds
Preprocessor visited node count: 1719/1000000
Post‐expand include size: 18736/2097152 bytes
Template argument size: 1789/2097152 bytes
Highest expansion depth: 13/40
Expensive parser function count: 3/500
Unstrip recursion depth: 1/20
Unstrip post‐expand size: 14154/5000000 bytes
Number of Wikibase entities loaded: 0/400
Lua time usage: 0.107/10.000 seconds
Lua memory usage: 3.33 MB/50 MB
-->
                        <!--
Transclusion expansion time report (%,ms,calls,template)
100.00%  423.690      1 -total
 26.69%  113.077      1 Template:Use_American_English
 21.93%   92.904      1 Template:Reflist
 19.39%   82.135      2 Template:Cite_book
 14.77%   62.599      1 Template:Short_description
  9.97%   42.248      1 Template:Expand_section
  7.81%   33.100      1 Template:Ambox
  7.76%   32.863      1 Template:Pagetype
  6.13%   25.975      1 Template:Citation_needed
  5.21%   22.058      1 Template:Fix
-->
                        <!-- Saved in parser cache with key enwiki:pcache:idhash:189682-0!canonical!math=5 and timestamp 20200224203346 and revision id 942460710
 -->
                    </div><noscript><img src="//en.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;" /></noscript>
                </div>
                <div class="printfooter">
                    Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&amp;oldid=942460710">https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&amp;oldid=942460710</a>"
                </div>
                <div id="catlinks" class="catlinks" data-mw="interface">
                    <div id="mw-normal-catlinks" class="mw-normal-catlinks">
                        <a href="/wiki/Help:Category" title="Help:Category">Categories</a>:
                        <ul>
                            <li>
                                <a href="/wiki/Category:Matrices" title="Category:Matrices">Matrices</a>
                            </li>
                        </ul>
                    </div>
                    <div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">
                        Hidden categories:
                        <ul>
                            <li>
                                <a href="/wiki/Category:Use_American_English_from_January_2019" title="Category:Use American English from January 2019">Use American English from January 2019</a>
                            </li>
                            <li>
                                <a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a>
                            </li>
                            <li>
                                <a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a>
                            </li>
                            <li>
                                <a href="/wiki/Category:Articles_to_be_expanded_from_February_2018" title="Category:Articles to be expanded from February 2018">Articles to be expanded from February 2018</a>
                            </li>
                            <li>
                                <a href="/wiki/Category:All_articles_to_be_expanded" title="Category:All articles to be expanded">All articles to be expanded</a>
                            </li>
                            <li>
                                <a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a>
                            </li>
                            <li>
                                <a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2019" title="Category:Articles with unsourced statements from September 2019">Articles with unsourced statements from September 2019</a>
                            </li>
                        </ul>
                    </div>
                </div>
                <div class="visualClear"></div>
            </div>
        </div>
        <div id="mw-data-after-content">
            <div class="read-more-container"></div>
        </div>
        <div id="mw-navigation">
            <h2>
                Navigation menu
            </h2>
            <div id="mw-head">
                <div id="p-personal" role="navigation" class="" aria-labelledby="p-personal-label">
                    <h3 id="p-personal-label">
                        Personal tools
                    </h3>
                    <ul>
                        <li id="pt-anonuserpage">Not logged in
                        </li>
                        <li id="pt-anontalk">
                            <a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a>
                        </li>
                        <li id="pt-anoncontribs">
                            <a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a>
                        </li>
                        <li id="pt-createaccount">
                            <a href="/w/index.php?title=Special:CreateAccount&amp;returnto=Hermitian+matrix" title="You are encouraged to create an account and log in; however, it is not mandatory">Create account</a>
                        </li>
                        <li id="pt-login">
                            <a href="/w/index.php?title=Special:UserLogin&amp;returnto=Hermitian+matrix" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o">Log in</a>
                        </li>
                    </ul>
                </div>
                <div id="left-navigation">
                    <div id="p-namespaces" role="navigation" class="vectorTabs" aria-labelledby="p-namespaces-label">
                        <h3 id="p-namespaces-label">
                            Namespaces
                        </h3>
                        <ul>
                            <li id="ca-nstab-main" class="selected">
                                <a href="/wiki/Hermitian_matrix" title="View the content page [c]" accesskey="c">Article</a>
                            </li>
                            <li id="ca-talk">
                                <a href="/wiki/Talk:Hermitian_matrix" rel="discussion" title="Discussion about the content page [t]" accesskey="t">Talk</a>
                            </li>
                        </ul>
                    </div>
                    <div id="p-variants" role="navigation" class="vectorMenu emptyPortlet" aria-labelledby="p-variants-label">
                        <input type="checkbox" class="vectorMenuCheckbox" aria-labelledby="p-variants-label" />
                        <h3 id="p-variants-label">
                            <span>Variants</span>
                        </h3>
                        <ul class="menu"></ul>
                    </div>
                </div>
                <div id="right-navigation">
                    <div id="p-views" role="navigation" class="vectorTabs" aria-labelledby="p-views-label">
                        <h3 id="p-views-label">
                            Views
                        </h3>
                        <ul>
                            <li id="ca-view" class="collapsible selected">
                                <a href="/wiki/Hermitian_matrix">Read</a>
                            </li>
                            <li id="ca-edit" class="collapsible">
                                <a href="/w/index.php?title=Hermitian_matrix&amp;action=edit" title="Edit this page [e]" accesskey="e">Edit</a>
                            </li>
                            <li id="ca-history" class="collapsible">
                                <a href="/w/index.php?title=Hermitian_matrix&amp;action=history" title="Past revisions of this page [h]" accesskey="h">View history</a>
                            </li>
                        </ul>
                    </div>
                    <div id="p-cactions" role="navigation" class="vectorMenu emptyPortlet" aria-labelledby="p-cactions-label">
                        <input type="checkbox" class="vectorMenuCheckbox" aria-labelledby="p-cactions-label" />
                        <h3 id="p-cactions-label">
                            <span>More</span>
                        </h3>
                        <ul class="menu"></ul>
                    </div>
                    <div id="p-search" role="search">
                        <h3>
                            <label for="searchInput">Search</label>
                        </h3>
                        <form action="/w/index.php" id="searchform" name="searchform">
                            <div id="simpleSearch">
                                <input type="search" name="search" placeholder="Search Wikipedia" title="Search Wikipedia [f]" accesskey="f" id="searchInput" /> <input type="hidden" value="Special:Search" name="title" /> <input type="submit" name="fulltext" value="Search" title="Search Wikipedia for this text" id="mw-searchButton" class="searchButton mw-fallbackSearchButton" /> <input type="submit" name="go" value="Go" title="Go to a page with this exact name if it exists" id="searchButton" class="searchButton" />
                            </div>
                        </form>
                    </div>
                </div>
            </div>
            <div id="mw-panel">
                <div id="p-logo" role="banner">
                    <a title="Visit the main page" class="mw-wiki-logo" href="/wiki/Main_Page"></a>
                </div>
                <div class="portal" role="navigation" id="p-navigation" aria-labelledby="p-navigation-label">
                    <h3 id="p-navigation-label">
                        Navigation
                    </h3>
                    <div class="body">
                        <ul>
                            <li id="n-mainpage-description">
                                <a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a>
                            </li>
                            <li id="n-contents">
                                <a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a>
                            </li>
                            <li id="n-featuredcontent">
                                <a href="/wiki/Wikipedia:Featured_content" title="Featured content – the best of Wikipedia">Featured content</a>
                            </li>
                            <li id="n-currentevents">
                                <a href="/wiki/Portal:Current_events" title="Find background information on current events">Current events</a>
                            </li>
                            <li id="n-randompage">
                                <a href="/wiki/Special:Random" title="Load a random article [x]" accesskey="x">Random article</a>
                            </li>
                            <li id="n-sitesupport">
                                <a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&amp;utm_medium=sidebar&amp;utm_campaign=C13_en.wikipedia.org&amp;uselang=en" title="Support us">Donate to Wikipedia</a>
                            </li>
                            <li id="n-shoplink">
                                <a href="//shop.wikimedia.org" title="Visit the Wikipedia store">Wikipedia store</a>
                            </li>
                        </ul>
                    </div>
                </div>
                <div class="portal" role="navigation" id="p-interaction" aria-labelledby="p-interaction-label">
                    <h3 id="p-interaction-label">
                        Interaction
                    </h3>
                    <div class="body">
                        <ul>
                            <li id="n-help">
                                <a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a>
                            </li>
                            <li id="n-aboutsite">
                                <a href="/wiki/Wikipedia:About" title="Find out about Wikipedia">About Wikipedia</a>
                            </li>
                            <li id="n-portal">
                                <a href="/wiki/Wikipedia:Community_portal" title="About the project, what you can do, where to find things">Community portal</a>
                            </li>
                            <li id="n-recentchanges">
                                <a href="/wiki/Special:RecentChanges" title="A list of recent changes in the wiki [r]" accesskey="r">Recent changes</a>
                            </li>
                            <li id="n-contactpage">
                                <a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact page</a>
                            </li>
                        </ul>
                    </div>
                </div>
                <div class="portal" role="navigation" id="p-tb" aria-labelledby="p-tb-label">
                    <h3 id="p-tb-label">
                        Tools
                    </h3>
                    <div class="body">
                        <ul>
                            <li id="t-whatlinkshere">
                                <a href="/wiki/Special:WhatLinksHere/Hermitian_matrix" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a>
                            </li>
                            <li id="t-recentchangeslinked">
                                <a href="/wiki/Special:RecentChangesLinked/Hermitian_matrix" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a>
                            </li>
                            <li id="t-upload">
                                <a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a>
                            </li>
                            <li id="t-specialpages">
                                <a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a>
                            </li>
                            <li id="t-permalink">
                                <a href="/w/index.php?title=Hermitian_matrix&amp;oldid=942460710" title="Permanent link to this revision of the page">Permanent link</a>
                            </li>
                            <li id="t-info">
                                <a href="/w/index.php?title=Hermitian_matrix&amp;action=info" title="More information about this page">Page information</a>
                            </li>
                            <li id="t-wikibase">
                                <a href="https://www.wikidata.org/wiki/Special:EntityPage/Q652941" title="Link to connected data repository item [g]" accesskey="g">Wikidata item</a>
                            </li>
                            <li id="t-cite">
                                <a href="/w/index.php?title=Special:CiteThisPage&amp;page=Hermitian_matrix&amp;id=942460710" title="Information on how to cite this page">Cite this page</a>
                            </li>
                        </ul>
                    </div>
                </div>
                <div class="portal" role="navigation" id="p-coll-print_export" aria-labelledby="p-coll-print_export-label">
                    <h3 id="p-coll-print_export-label">
                        Print/export
                    </h3>
                    <div class="body">
                        <ul>
                            <li id="coll-create_a_book">
                                <a href="/w/index.php?title=Special:Book&amp;bookcmd=book_creator&amp;referer=Hermitian+matrix">Create a book</a>
                            </li>
                            <li id="coll-download-as-rl">
                                <a href="/w/index.php?title=Special:ElectronPdf&amp;page=Hermitian+matrix&amp;action=show-download-screen">Download as PDF</a>
                            </li>
                            <li id="t-print">
                                <a href="/w/index.php?title=Hermitian_matrix&amp;printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a>
                            </li>
                        </ul>
                    </div>
                </div>
                <div class="portal" role="navigation" id="p-lang" aria-labelledby="p-lang-label">
                    <h3 id="p-lang-label">
                        Languages
                    </h3>
                    <div class="body">
                        <ul>
                            <li class="interlanguage-link interwiki-ar">
                                <a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D9%87%D9%8A%D8%B1%D9%85%D9%8A%D8%AA%D9%8A%D8%A9" title="مصفوفة هيرميتية – Arabic" lang="ar" hreflang="ar" class="interlanguage-link-target" xml:lang="ar">العربية</a>
                            </li>
                            <li class="interlanguage-link interwiki-bn">
                                <a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A6%BE%E0%A6%B0%E0%A6%AE%E0%A6%BF%E0%A6%B6%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8" title="হারমিশিয়ান – Bangla" lang="bn" hreflang="bn" class="interlanguage-link-target" xml:lang="bn">বাংলা</a>
                            </li>
                            <li class="interlanguage-link interwiki-be">
                                <a href="https://be.wikipedia.org/wiki/%D0%AD%D1%80%D0%BC%D1%96%D1%82%D0%B0%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Эрмітава матрыца – Belarusian" lang="be" hreflang="be" class="interlanguage-link-target" xml:lang="be">Беларуская</a>
                            </li>
                            <li class="interlanguage-link interwiki-ca">
                                <a href="https://ca.wikipedia.org/wiki/Matriu_herm%C3%ADtica" title="Matriu hermítica – Catalan" lang="ca" hreflang="ca" class="interlanguage-link-target" xml:lang="ca">Català</a>
                            </li>
                            <li class="interlanguage-link interwiki-cs">
                                <a href="https://cs.wikipedia.org/wiki/Hermitovsk%C3%A1_matice" title="Hermitovská matice – Czech" lang="cs" hreflang="cs" class="interlanguage-link-target" xml:lang="cs">Čeština</a>
                            </li>
                            <li class="interlanguage-link interwiki-de">
                                <a href="https://de.wikipedia.org/wiki/Hermitesche_Matrix" title="Hermitesche Matrix – German" lang="de" hreflang="de" class="interlanguage-link-target" xml:lang="de">Deutsch</a>
                            </li>
                            <li class="interlanguage-link interwiki-et">
                                <a href="https://et.wikipedia.org/wiki/Hermiitiline_maatriks" title="Hermiitiline maatriks – Estonian" lang="et" hreflang="et" class="interlanguage-link-target" xml:lang="et">Eesti</a>
                            </li>
                            <li class="interlanguage-link interwiki-es">
                                <a href="https://es.wikipedia.org/wiki/Matriz_hermitiana" title="Matriz hermitiana – Spanish" lang="es" hreflang="es" class="interlanguage-link-target" xml:lang="es">Español</a>
                            </li>
                            <li class="interlanguage-link interwiki-eo">
                                <a href="https://eo.wikipedia.org/wiki/Memadjunkta_matrico" title="Memadjunkta matrico – Esperanto" lang="eo" hreflang="eo" class="interlanguage-link-target" xml:lang="eo">Esperanto</a>
                            </li>
                            <li class="interlanguage-link interwiki-fa">
                                <a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%D9%87%D8%B1%D9%85%DB%8C%D8%AA%DB%8C" title="ماتریس هرمیتی – Persian" lang="fa" hreflang="fa" class="interlanguage-link-target" xml:lang="fa">فارسی</a>
                            </li>
                            <li class="interlanguage-link interwiki-fr">
                                <a href="https://fr.wikipedia.org/wiki/Hermitien" title="Hermitien – French" lang="fr" hreflang="fr" class="interlanguage-link-target" xml:lang="fr">Français</a>
                            </li>
                            <li class="interlanguage-link interwiki-ko">
                                <a href="https://ko.wikipedia.org/wiki/%EC%97%90%EB%A5%B4%EB%AF%B8%ED%8A%B8_%ED%96%89%EB%A0%AC" title="에르미트 행렬 – Korean" lang="ko" hreflang="ko" class="interlanguage-link-target" xml:lang="ko">한국어</a>
                            </li>
                            <li class="interlanguage-link interwiki-it">
                                <a href="https://it.wikipedia.org/wiki/Matrice_hermitiana" title="Matrice hermitiana – Italian" lang="it" hreflang="it" class="interlanguage-link-target" xml:lang="it">Italiano</a>
                            </li>
                            <li class="interlanguage-link interwiki-lv">
                                <a href="https://lv.wikipedia.org/wiki/Ermita_matrica" title="Ermita matrica – Latvian" lang="lv" hreflang="lv" class="interlanguage-link-target" xml:lang="lv">Latviešu</a>
                            </li>
                            <li class="interlanguage-link interwiki-lt">
                                <a href="https://lt.wikipedia.org/wiki/Ermito_matrica" title="Ermito matrica – Lithuanian" lang="lt" hreflang="lt" class="interlanguage-link-target" xml:lang="lt">Lietuvių</a>
                            </li>
                            <li class="interlanguage-link interwiki-hu">
                                <a href="https://hu.wikipedia.org/wiki/Hermitikus_m%C3%A1trix" title="Hermitikus mátrix – Hungarian" lang="hu" hreflang="hu" class="interlanguage-link-target" xml:lang="hu">Magyar</a>
                            </li>
                            <li class="interlanguage-link interwiki-nl">
                                <a href="https://nl.wikipedia.org/wiki/Hermitische_matrix" title="Hermitische matrix – Dutch" lang="nl" hreflang="nl" class="interlanguage-link-target" xml:lang="nl">Nederlands</a>
                            </li>
                            <li class="interlanguage-link interwiki-ja">
                                <a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%AB%E3%83%9F%E3%83%BC%E3%83%88%E8%A1%8C%E5%88%97" title="エルミート行列 – Japanese" lang="ja" hreflang="ja" class="interlanguage-link-target" xml:lang="ja">日本語</a>
                            </li>
                            <li class="interlanguage-link interwiki-pl">
                                <a href="https://pl.wikipedia.org/wiki/Macierz_hermitowska" title="Macierz hermitowska – Polish" lang="pl" hreflang="pl" class="interlanguage-link-target" xml:lang="pl">Polski</a>
                            </li>
                            <li class="interlanguage-link interwiki-pt">
                                <a href="https://pt.wikipedia.org/wiki/Matriz_hermitiana" title="Matriz hermitiana – Portuguese" lang="pt" hreflang="pt" class="interlanguage-link-target" xml:lang="pt">Português</a>
                            </li>
                            <li class="interlanguage-link interwiki-ru">
                                <a href="https://ru.wikipedia.org/wiki/%D0%AD%D1%80%D0%BC%D0%B8%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Эрмитова матрица – Russian" lang="ru" hreflang="ru" class="interlanguage-link-target" xml:lang="ru">Русский</a>
                            </li>
                            <li class="interlanguage-link interwiki-sl">
                                <a href="https://sl.wikipedia.org/wiki/Hermitska_matrika" title="Hermitska matrika – Slovenian" lang="sl" hreflang="sl" class="interlanguage-link-target" xml:lang="sl">Slovenščina</a>
                            </li>
                            <li class="interlanguage-link interwiki-sr">
                                <a href="https://sr.wikipedia.org/wiki/%D0%95%D1%80%D0%BC%D0%B8%D1%82%D1%81%D0%BA%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Ермитска матрица – Serbian" lang="sr" hreflang="sr" class="interlanguage-link-target" xml:lang="sr">Српски / srpski</a>
                            </li>
                            <li class="interlanguage-link interwiki-fi">
                                <a href="https://fi.wikipedia.org/wiki/Hermiittinen_matriisi" title="Hermiittinen matriisi – Finnish" lang="fi" hreflang="fi" class="interlanguage-link-target" xml:lang="fi">Suomi</a>
                            </li>
                            <li class="interlanguage-link interwiki-sv">
                                <a href="https://sv.wikipedia.org/wiki/Hermitesk_matris" title="Hermitesk matris – Swedish" lang="sv" hreflang="sv" class="interlanguage-link-target" xml:lang="sv">Svenska</a>
                            </li>
                            <li class="interlanguage-link interwiki-ta">
                                <a href="https://ta.wikipedia.org/wiki/%E0%AE%B9%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AF%88%E0%AE%9F%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="ஹெர்மைட் அணி – Tamil" lang="ta" hreflang="ta" class="interlanguage-link-target" xml:lang="ta">தமிழ்</a>
                            </li>
                            <li class="interlanguage-link interwiki-th">
                                <a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B9%81%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%A1%E0%B8%B4%E0%B8%95" title="เมทริกซ์แอร์มิต – Thai" lang="th" hreflang="th" class="interlanguage-link-target" xml:lang="th">ไทย</a>
                            </li>
                            <li class="interlanguage-link interwiki-tr">
                                <a href="https://tr.wikipedia.org/wiki/Hermisyen_matris" title="Hermisyen matris – Turkish" lang="tr" hreflang="tr" class="interlanguage-link-target" xml:lang="tr">Türkçe</a>
                            </li>
                            <li class="interlanguage-link interwiki-uk">
                                <a href="https://uk.wikipedia.org/wiki/%D0%95%D1%80%D0%BC%D1%96%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F" title="Ермітова матриця – Ukrainian" lang="uk" hreflang="uk" class="interlanguage-link-target" xml:lang="uk">Українська</a>
                            </li>
                            <li class="interlanguage-link interwiki-zh">
                                <a href="https://zh.wikipedia.org/wiki/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="埃尔米特矩阵 – Chinese" lang="zh" hreflang="zh" class="interlanguage-link-target" xml:lang="zh">中文</a>
                            </li>
                        </ul>
                        <div class="after-portlet after-portlet-lang">
                            <span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q652941#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span>
                        </div>
                    </div>
                </div>
            </div>
        </div>
        <div id="footer" role="contentinfo">
            <ul id="footer-info" class="">
                <li id="footer-info-lastmod">This page was last edited on 24 February 2020, at 20:33<span class="anonymous-show">&#160;(UTC)</span>.
                </li>
                <li id="footer-info-copyright">Text is available under the <a rel="license" href="//en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_License">Creative Commons Attribution-ShareAlike License</a><a rel="license" href="//creativecommons.org/licenses/by-sa/3.0/" style="display:none;"></a>; additional terms may apply. By using this site, you agree to the <a href="//foundation.wikimedia.org/wiki/Terms_of_Use">Terms of Use</a> and <a href="//foundation.wikimedia.org/wiki/Privacy_policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a href="//www.wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.
                </li>
            </ul>
            <ul id="footer-places" class="">
                <li id="footer-places-privacy">
                    <a href="https://foundation.wikimedia.org/wiki/Privacy_policy" class="extiw" title="wmf:Privacy policy">Privacy policy</a>
                </li>
                <li id="footer-places-about">
                    <a href="/wiki/Wikipedia:About" title="Wikipedia:About">About Wikipedia</a>
                </li>
                <li id="footer-places-disclaimer">
                    <a href="/wiki/Wikipedia:General_disclaimer" title="Wikipedia:General disclaimer">Disclaimers</a>
                </li>
                <li id="footer-places-contact">
                    <a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a>
                </li>
                <li id="footer-places-developers">
                    <a href="https://www.mediawiki.org/wiki/Special:MyLanguage/How_to_contribute">Developers</a>
                </li>
                <li id="footer-places-statslink">
                    <a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a>
                </li>
                <li id="footer-places-cookiestatement">
                    <a href="https://foundation.wikimedia.org/wiki/Cookie_statement">Cookie statement</a>
                </li>
                <li id="footer-places-mobileview">
                    <a href="//en.m.wikipedia.org/w/index.php?title=Hermitian_matrix&amp;mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a>
                </li>
            </ul>
            <ul id="footer-icons" class="noprint">
                <li id="footer-copyrightico">
                    <a href="https://wikimediafoundation.org/"><img src="/static/images/wikimedia-button.png" srcset="/static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x" width="88" height="31" alt="Wikimedia Foundation" /></a>
                </li>
                <li id="footer-poweredbyico">
                    <a href="https://www.mediawiki.org/"><img src="/static/images/poweredby_mediawiki_88x31.png" alt="Powered by MediaWiki" srcset="/static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x" width="88" height="31" /></a>
                </li>
            </ul>
            <div style="clear: both;"></div>
        </div>
        <script>
        <![CDATA[
        (RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgPageParseReport":{"limitreport":{"cputime":"0.344","walltime":"0.623","ppvisitednodes":{"value":1719,"limit":1000000},"postexpandincludesize":{"value":18736,"limit":2097152},"templateargumentsize":{"value":1789,"limit":2097152},"expansiondepth":{"value":13,"limit":40},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":14154,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00%  423.690      1 -total"," 26.69%  113.077      1 Template:Use_American_English"," 21.93%   92.904      1 Template:Reflist"," 19.39%   82.135      2 Template:Cite_book"," 14.77%   62.599      1 Template:Short_description","  9.97%   42.248      1 Template:Expand_section","  7.81%   33.100      1 Template:Ambox","  7.76%   32.863      1 Template:Pagetype","  6.13%   25.975      1 Template:Citation_needed","  5.21%   22.058      1 Template:Fix"]},"scribunto":{"limitreport-timeusage":{"value":"0.107","limit":"10.000"},"limitreport-memusage":{"value":3494345,"limit":52428800}},"cachereport":{"origin":"mw1375","timestamp":"20200224203345","ttl":2592000,"transientcontent":false}}});});
        ]]>
        </script> 
        <script type="application/ld+json">
        <![CDATA[
        {"@context":"https:\/\/schema.org","@type":"Article","name":"Hermitian matrix","url":"https:\/\/en.wikipedia.org\/wiki\/Hermitian_matrix","sameAs":"http:\/\/www.wikidata.org\/entity\/Q652941","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q652941","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-02-28T21:51:08Z","dateModified":"2020-02-24T20:33:46Z","headline":"matrix equal to its conjugate-transpose"}
        ]]>
        </script> 
        <script>
        <![CDATA[
        (RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgBackendResponseTime":114,"wgHostname":"mw1248"});});
        ]]>
        </script>
    </body>
</html>
